On an inequality of Sagher and Zhou concerning Stein's lemma

0334985 - MÚ 2010 RIV ES eng J - Journal Article
Announi, M. - Grafakos, L. - Honzík, Petr
On an inequality of Sagher and Zhou concerning Stein's lemma.
[O nerovnosti Saghrera a Zhoua vztahující se k Steinovu lemmatu.]
Collectanea Mathematica. Roč. 60, č. 3 (2009), s. 297-306. ISSN 0010-0757. E-ISSN 2038-4815
Institutional research plan: CEZ:AV0Z10190503
Keywords : lacunary series * sequences * Rademacher functions
Subject RIV: BA - General Mathematics
Impact factor: 0.389, year: 2009

We provide two alternative proofs of the following formulation of Stein's lemma obtained by Sagher and Zhou [6]: there exists a constant A > 0 such that for any measurable set E subset of [0, 1], vertical bar E vertical bar not equal 0, there is an integer N that depends only on E such that for any square-summable real-valued sequence {c(k)}(k=0)(infinity) we have: A.Sigma(k > N)vertical bar c(k)vertical bar(2) <= sup(I) inf(a is an element of R) 1/vertical bar I vertical bar integral(I boolean AND E) vertical bar f(t) - a vertical bar(2) dt, (1)where the supremum is taken over all dyadic intervals I and f(t) = Sigma(infinity)(k=0)c(k)(sic)(k)(t), where (sic)(k) denotes the kth Rademacher function. The first proof does not rely on Khintchine's inequality while the second is succinct and applies to general lacunary Walsh series.

Ukazeme dva nové způsoby, jak dokázat nerovnost Saghera a Zhoua.
Permanent Link: http://hdl.handle.net/11104/0179583